GCSE Maths Practice: conditional-probability

Question 5 of 13

This question focuses on conditional probability by restricting the sample space based on given information.

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\( \begin{array}{l}\text{A bag contains 5 red balls, 3 green balls, and 2 blue balls.} \\ \text{One ball is drawn at random.} \\ \text{What is the probability the ball is green, given that it is not red?}\end{array} \)

Choose one option:

Remove the excluded outcomes first, then calculate probability from what remains.

Conditional Probability by Restricting the Sample Space

Conditional probability is used when additional information is given that changes which outcomes are possible. In this type of question, the condition does not involve a second draw or replacement. Instead, it tells us something about the outcome that has already happened, allowing us to eliminate certain possibilities.

Here, the key phrase is "given that it is not red". This information allows us to immediately remove all red balls from the sample space. Once those outcomes are excluded, we calculate probability using only the remaining items.

Understanding the Core Idea

Probability is always calculated as:

Number of favourable outcomes ÷ total number of possible outcomes

When a condition is applied, the total number of possible outcomes may shrink. This is what makes the probability conditional. We are no longer working with the original bag, but with a smaller, restricted version of it.

Step-by-Step Method

  1. Read the condition carefully.
  2. Remove any outcomes that are no longer possible.
  3. Count how many outcomes remain in total.
  4. Count how many of those outcomes are favourable.
  5. Form the probability using the restricted total.

Worked Example 1

A box contains 4 apples, 3 pears, and 3 oranges. One fruit is chosen at random. What is the probability the fruit is a pear, given that it is not an apple?

Answer: Removing apples leaves 3 pears and 3 oranges, giving 6 fruits. The probability of pear is \(\frac{3}{6} = \frac{1}{2}\).

Worked Example 2

A jar contains 6 black beads, 2 white beads, and 2 yellow beads. One bead is selected. What is the probability it is yellow, given that it is not black?

Answer: Excluding black beads leaves 2 white and 2 yellow beads, giving 4 beads. The probability of yellow is \(\frac{2}{4} = \frac{1}{2}\).

Common Mistakes to Avoid

  • Using the original total instead of the restricted total.
  • Forgetting to remove excluded outcomes.
  • Thinking a second event is involved when there is only one.
  • Confusing this with questions involving replacement.

Real-Life Application

This type of reasoning is common in real life. For example, if you know a customer did not choose a vegetarian option, you can restrict your analysis to the remaining meal choices. Conditional probability allows you to update decisions based on new information.

Frequently Asked Questions

Is this still conditional probability if there is only one draw?
Yes. The condition changes the sample space, which makes the probability conditional.

Do I need a formula?
No. At GCSE Foundation level, careful counting is usually sufficient.

How can I spot these questions?
Look for phrases like “given that”, “knowing that”, or “it is not”.

Study Tip

Before calculating anything, rewrite the situation using only the outcomes that are still possible. This makes the probability much easier to see.

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